Histograms and Ogive ICSE Class 10: Mode, Median, Quartiles and IQR from Graphs

What Histograms and Ogive is about

Weightage: CISCE publishes no chapter or unit weightage for ICSE Mathematics. Histograms and Ogive is in the 6. Statistics (b) part of the 2027 syllabus and can be asked in both Section A and Section B of the 80-mark paper.Source: CISCE ICSE Mathematics syllabus, examination year 2027

Key concepts

Drawing a histogram

A histogram is a set of adjacent rectangles (no gaps). With equal class widths:

• x-axis: class boundaries, with uniform spacing; y-axis: frequency.
• Each bar's width is the class width and its height is the frequency.
• If the x-axis does not start at 0, show a kink (a small zig-zag) near the origin.
• Use the scale given in the question on both axes and label both axes.

For discontinuous (inclusive) classes like 11–20, 21–30, first convert to boundaries 10.5–20.5, 20.5–30.5, …

Worked example

Which boundaries do you mark on the x-axis for the classes 11–20, 21–30, 31–40 and 41–50?

Try it first, then show the solutionSolution

1. The gap between 20 and 21 is 1, so subtract 0.5 from each lower limit and add 0.5 to each upper limit.
2. Boundaries: 10.5, 20.5, 30.5, 40.5, 50.5 (with a kink before 10.5).

Answer: 10.5, 20.5, 30.5, 40.5, 50.5

Mode from a histogram

1. Find the tallest bar (the modal class).
2. Join the top-right corner of the tallest bar to the top-right corner of the bar before it, and the top-left corner of the tallest bar to the top-left corner of the bar after it. The two lines cross inside the tallest bar.
3. From the crossing point, draw a perpendicular to the x-axis.
4. The foot of the perpendicular gives the mode.

The mode is a value on the x-axis, not the frequency and not the class.

Worked example

For classes 10–20, 20–30, 30–40, 40–50, 50–60 with frequencies 6, 10, 18, 12, 4, what mode should the histogram show?

Try it first, then show the solutionSolution

1. Tallest bar: 30–40 (18). Neighbours: 10 (before) and 12 (after).
2. The crossing lines give the same value as $30+\frac{18-10}{(18-10)+(18-12)}\times 10$.
3. $=30+\frac{8}{14}\times 10\approx 35.7$.

Answer: Mode ≈ 35.7 (modal class 30–40)

Cumulative frequency (less than)

The less than cumulative frequency of a class is the number of observations below its upper boundary: add the frequencies down the table.

The last cumulative frequency must equal the total $N$; check it.

Worked example

Make the less than cumulative frequency table for marks: 0–10: 5, 10–20: 9, 20–30: 14, 30–40: 20, 40–50: 16, 50–60: 10, 60–70: 6.

Try it first, then show the solutionSolution

1. Less than 10: 5; less than 20: 14; less than 30: 28; less than 40: 48.
2. Less than 50: 64; less than 60: 74; less than 70: 80.
3. $N=80$ ✓

Answer: 5, 14, 28, 48, 64, 74, 80

Drawing a less than ogive

1. Plot each cumulative frequency against the upper class boundary (not the class mark, not the lower limit).
2. Also plot the starting point (lowest lower boundary, 0).
3. Join the points with a smooth free-hand curve (not a ruler).
4. Use the given scale; show a kink if an axis does not start at 0; label the axes.

The ogive rises from left to right, steepest where the frequencies are largest.

Worked example

List the points to plot for the marks table above.

Try it first, then show the solutionSolution

1. Start: (0, 0).
2. (10, 5), (20, 14), (30, 28), (40, 48), (50, 64), (60, 74), (70, 80).

Answer: (0, 0), (10, 5), (20, 14), (30, 28), (40, 48), (50, 64), (60, 74), (70, 80)

Median and quartiles from the ogive

For $N$ observations, find the positions on the cumulative frequency axis:

• Median: $\frac{N}{2}$th observation.
• Lower quartile $Q_1$: $\frac{N}{4}$th.
• Upper quartile $Q_3$: $\frac{3N}{4}$th.

From the position on the y-axis, draw a horizontal line to the curve, then a perpendicular down to the x-axis; read the value. Show these lines on the graph.

Because the ogive is drawn free-hand, answers are approximate; a straight-line estimate inside the class gives a good check.

Worked example

For the 80 marks above, estimate the median, $Q_1$ and $Q_3$.

Try it first, then show the solutionSolution

1. Median: 40th. It lies in 30–40 (c.f. 28 to 48): $30+\frac{40-28}{20}\times 10=36$.
2. $Q_1$: 20th, in 20–30: $20+\frac{20-14}{14}\times 10\approx 24.3$.
3. $Q_3$: 60th, in 40–50: $40+\frac{60-48}{16}\times 10=47.5$.

Answer: Median ≈ 36, Q₁ ≈ 24.3, Q₃ ≈ 47.5

Interquartile range and reading counts

Interquartile range (IQR) $=Q_3-Q_1$: the spread of the middle half of the data. A smaller IQR means the data are more consistent.

To find how many observations lie below a value $x$: draw a perpendicular up from $x$ to the curve, then across to the y-axis. 'More than $x$' $=N-$ 'less than $x$'. 'Between $a$ and $b$' $=$ (less than $b$) $-$ (less than $a$).

Worked example

For the marks ogive, find the IQR and the number of students scoring more than 55.

Try it first, then show the solutionSolution

1. IQR $\approx 47.5-24.3=23.2$.
2. Less than 55: about $64+\frac{5}{10}\times 10=69$.
3. More than 55: $80-69=11$.

Answer: IQR ≈ 23.2; about 11 students

Maths in real life: Histograms and Ogive

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Histogram and Mode
• Drawing a Less Than Ogive
• Median and Quartiles from an Ogive

Histograms and Ogive formula sheet

Median position (ogive)

$\frac{N}{2}\text{th observation}$

Read across to the curve, then down.

Quartile positions

$Q_1:\frac{N}{4}\text{th},Q_3:\frac{3N}{4}\text{th}$

N = total frequency.

Interquartile range

$\text{IQR}=Q_3-Q_1$

Spread of the middle 50%.

Mode from histogram (check)

$L+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h$

Matches the crossing-lines construction; f₁ modal, f₀ before, f₂ after.

Class boundaries for inclusive classes

$\text{lower}-0.5,\text{upper}+0.5$

For classes like 11–20, 21–30.

More than / between

$N-\text{c.f.}(x),\text{c.f.}(b)-\text{c.f.}(a)$

Read c.f. values off the ogive.

Histograms and Ogive: diagrams to practise

Practise each one on paper: draw it from the checklist, then tick off every label.

How the board asks Histograms and Ogive

The question patterns to expect, and the method that works for each.

Tricks to remember Histograms and Ogive

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Ogive = Upper

  'O' for ogive, 'U' for upper: less than ogives use upper boundaries.

  Use it when: Plotting the right x-values.

• Cross the tallest

  In the tallest bar, the two lines make an X; drop from the X to find the mode.

  Use it when: Mode from a histogram.

• Quarter, half, three-quarters

  Q₁ at N/4, median at N/2, Q₃ at 3N/4.

  Use it when: Ogive read-offs.

• Kink when not zero

  Axis starts away from zero? Draw the zig-zag.

  Use it when: Avoiding the scale deduction.

Histograms and Ogive: board-level solved questions

Ordered from board level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section B style)

Read the situation, then answer every part. In the board paper this is a 10-mark question split into 3 + 3 + 4 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Histograms and Ogive

The slips examiners see most often, and what to write instead.

• ✗Mistake: Plotting against the wrong x-values

  CISCE examiners' comments, 2024: many candidates plotted the ogive against lower boundaries or class marks instead of upper class boundaries; some drew a histogram instead of an ogive.

  ✓Do this: Less than ogive: c.f. against the upper boundary, starting from (lowest boundary, 0).

  Usually costs: Most of the graph marks.

• ✗Mistake: Ruler instead of a free-hand curve

  CISCE examiners' comments, 2024: candidates joined the ogive points with a ruler.

  ✓Do this: Join with one smooth free-hand curve.

  Usually costs: 1 mark.

• ✗Mistake: No kink, wrong scale, uneven spacing

  CISCE examiners' comments, 2024 and 2025: no kink when the axis did not start at zero, the given scale not used, x-axis labelled with class intervals instead of boundaries, and non-uniform spacing.

  ✓Do this: Use exactly the scale given; mark boundaries at equal spacing; draw the kink.

  Usually costs: 1 mark each.

• ✗Mistake: No read-off lines

  CISCE examiners' comments, 2024: candidates did not draw perpendicular lines to read values from the graph.

  ✓Do this: Draw the horizontal and vertical lines for every value you read, and write the value beside them.

  Usually costs: 1 mark per value.

• ✗Mistake: Value confused with class

  CISCE examiners' comments, 2024 and 2025: candidates gave the mode when the modal class was asked (and the reverse), wrote the frequency 12 as the mode, gave the median value instead of the median class, and wrote ₹2–₹4 instead of ₹2000–₹4000.

  ✓Do this: Class = an interval in full units; mode, median, quartile = a single value on the x-axis.

  Usually costs: 1 mark.

• ✗Mistake: Misreading the scale

  CISCE examiners' comments, 2025: candidates misread a given graph, for example writing 80 instead of 75 for the total number of workers, and could not read the mode accurately from a correct graph.

  ✓Do this: Work out what one small square represents before reading any value.

  Usually costs: 1 mark per misread.

Practice questions on Histograms and Ogive

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQBoard level

   In a histogram with equal class widths, the height of each bar represents:
   (a) the class mark
   (b) the frequency
   (c) the cumulative frequency
   (d) the class width

   Hint

   Bars stand on the classes.

   Answer

   (b)

2. Q2 · 1 mark · MCQBoard level

   For 80 observations, the upper quartile is read at the:
   (a) 20th observation
   (b) 40th observation
   (c) 60th observation
   (d) 75th observation

   Hint

   3N/4.

   Answer

   (c)

3. Q3 · 1 mark · MCQHarder board level

   If $Q_1=24$ and $Q_3=47$, the interquartile range is:
   (a) 11.5
   (b) 23
   (c) 35.5
   (d) 71

   Hint

   Q₃ − Q₁.

   Answer

   (b)

4. Q4 · 1 mark · Assertion-reasonBoard level

   Assertion (A): The first point plotted for a less than ogive has cumulative frequency 0 at the lower boundary of the first class.
   Reason (R): No observation is less than the lower boundary of the first class.
   (a) (A) is true and (R) is false.
   (b) (A) is false and (R) is true.
   (c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
   (d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).

   Hint

   Count below the first boundary.

   Answer

   (c)

5. Q5 · 4 marks · Short answerHarder board level

   The less than cumulative frequencies for classes 0–5, 5–10, 10–15, 15–20, 20–25, 25–30 are 3, 10, 22, 37, 45, 50. Write the frequency table, and state the modal class and the median class.

   Hint

   Subtract consecutive c.f. values.

   Answer

   Frequencies 3, 7, 12, 15, 8, 5. Modal class 15–20 (15). Median: 25th observation, c.f. passes 25 in 15–20.

6. Q6 · 4 marks · Short answerHarder board level

   For 0–10: 4, 10–20: 10, 20–30: 6, 30–40: 3, explain how the histogram gives the mode, and estimate it.

   Hint

   Neighbours of the tallest bar are 4 and 6.

   Answer

   Tallest bar 10–20; join its top corners to the opposite top corners of the neighbouring bars; drop a perpendicular from the crossing point to the x-axis. Mode $\approx 10+\frac{10-4}{(10-4)+(10-6)}\times 10=16$.

7. Q7 · 3 marks · Long answerBoard level

   Convert the classes 1–10, 11–20, 21–30, 31–40 into class boundaries and write the less than cumulative frequency table for frequencies 6, 11, 9, 4.

   Hint

   Subtract and add 0.5.

   Answer

   0.5–10.5, 10.5–20.5, 20.5–30.5, 30.5–40.5. c.f.: less than 10.5: 6; 20.5: 17; 30.5: 26; 40.5: 30.

8. Q8 · 4 marks · Long answerHarder board level

   Draw a less than ogive for the heights of 50 students: 140–145: 4, 145–150: 10, 150–155: 16, 155–160: 12, 160–165: 6, 165–170: 2. Estimate the median, the IQR, and the number of students taller than 158 cm.

   Hint

   c.f.: 4, 14, 30, 42, 48, 50.

   Answer

   Median (25th) $\approx 153.4$ cm. $Q_1$ (12.5th) $\approx 149.3$, $Q_3$ (37.5th) $\approx 158.1$, IQR $\approx 8.9$ cm. Less than 158 $\approx 37$, so about 13 students are taller.

9. Q9 · 4 marks · Long answerHOTS / competency

   Two classes took the same test. Class X: median 50, $Q_1=42$, $Q_3=58$. Class Y: median 50, $Q_1=35$, $Q_3=66$. Which class performed more consistently? Explain, and say what the ogive of class X looks like compared with class Y near the median.

   Hint

   Compare IQRs; steeper means bunched.

   Answer

   IQR: X $=16$, Y $=31$. Class X is more consistent: the middle half of its marks lie in a narrower band. X's ogive rises more steeply around 50, because the same number of students fall in a shorter interval.

10. Q10 · 4 marks · Long answerHarder board level

    From the ogive of the 80 students' marks (c.f. 5, 14, 28, 48, 64, 74, 80 at 10, 20, …, 70), how many students scored at least 30 but less than 50? What percentage is this?

    Hint

    c.f.(50) − c.f.(30).

    Answer

    $64-28=36$ students; $\frac{36}{80}\times 100=45\%$.

11. Q11 · 1 mark · MCQHOTS / competency

    In a distribution, 25% of the observations are below 40 and 75% are below 65. The interquartile range is:
    (a) 25
    (b) 40
    (c) 52.5
    (d) 105

    Hint

    These are Q₁ and Q₃.

    Answer

    (a)

12. Q12 · 5 marks · Long answerHOTS / competency

    The distances (km) 60 students travel to school are: 0–2: 6, 2–4: 14, 4–6: 18, 6–8: 12, 8–10: 6, 10–12: 4. Draw an ogive (2 cm = 2 km, 2 cm = 10 students) and estimate (a) the median, (b) the IQR, (c) the percentage who travel more than 7 km.

    Hint

    c.f.: 6, 20, 38, 50, 56, 60.

    Answer

    (a) Median (30th) $\approx 5.1$ km. (b) $Q_1$ (15th) $\approx 3.3$, $Q_3$ (45th) $\approx 7.2$; IQR $\approx 3.9$ km. (c) Less than 7 km $\approx 44$; more than 7 km $\approx 16$, i.e. about 26.7%.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.Less than ogive: x-values plotted?Show answer

   Upper class boundaries.

2. 2.First point of a less than ogive?Show answer

   (Lowest lower boundary, 0).

3. 3.Position of Q₃ for N observations?Show answer

   3N/4 th

4. 4.Interquartile range?Show answer

   Q₃ − Q₁

5. 5.How is the mode found from a histogram?Show answer

   Crossing lines in the tallest bar, then a perpendicular to the x-axis.

6. 6.What does a kink show?Show answer

   That the axis does not start at 0.

7. 7.Boundaries for 21–30?Show answer

   20.5–30.5

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Do you use upper boundaries for the ogive?
• ☐Do you start the ogive at c.f. 0?
• ☐Do you join free-hand?
• ☐Do you show the kink and the given scale?
• ☐Do you draw read-off lines?
• ☐Do you give classes in full and values as single numbers?

Histograms and Ogive: frequently asked questions